Structural properties of the Caenorhabditis elegans neuronal network

Communities in C. elegans connectome through the prism of non-backtracking walks

The fundamental relationship between the mesoscopic structure of neuronal circuits and organismic functions they subserve is one of the major challenges in contemporary neuroscience. Formation of structurally connected modules of neurons enacts the conversion from single-cell firing to large-scale behaviour of an organism, highlighting the importance of their accurate profiling in the data. While connectomes are typically characterized by significant sparsity of neuronal connections, recent advances in network theory and machine learning have revealed fundamental limitations of traditionally used community detection approaches in cases where the network is sparse. Here we studied the optimal community structure in the structural connectome of Caenorhabditis elegans, for which we exploited a non-conventional approach that is based on non-backtracking random walks, virtually eliminating the sparsity issue. In full agreement with the previous asymptotic results, we demonstrated that non-backtracking walks resolve the ground truth annotation into clusters on stochastic block models (SBM) with the size and density of the connectome better than the spectral methods related to simple random walks. Based on the cluster detectability threshold, we determined that the optimal number of modules in a recently mapped connectome of C. elegans is 10, which precisely corresponds to the number of isolated eigenvalues in the spectrum of the non-backtracking flow matrix. The discovered communities have a clear interpretation in terms of their functional role, which allows one to discern three structural compartments in the worm: the Worm Brain (WB), the Worm Movement Controller (WMC), and the Worm Information Flow Connector (WIFC). Broadly, our work provides a robust network-based framework to reveal mesoscopic structures in sparse connectomic datasets, paving way to further investigation of connectome mechanisms for different functions.

Mechanical intelligence simplifies control in terrestrial limbless locomotion

Limbless locomotors, from microscopic worms to macroscopic snakes, traverse complex, heterogeneous natural environments typically using undulatory body wave propagation. Theoretical and robophysical models typically emphasize body kinematics and active neural/electronic control. However, we contend that because such approaches often neglect the role of passive, mechanically controlled processes (those involving "mechanical intelligence"), they fail to reproduce the performance of even the simplest organisms. To uncover principles of how mechanical intelligence aids limbless locomotion in heterogeneous terradynamic regimes, here we conduct a comparative study of locomotion in a model of heterogeneous terrain (lattices of rigid posts). We used a model biological system, the highly studied nematode worm Caenorhabditis elegans, and a robophysical device whose bilateral actuator morphology models that of limbless organisms across scales. The robot's kinematics quantitatively reproduced the performance of the nematodes with purely open-loop control; mechanical intelligence simplified control of obstacle navigation and exploitation by reducing the need for active sensing and feedback. An active behavior observed in C. elegans, undulatory wave reversal upon head collisions, robustified locomotion via exploitation of the systems' mechanical intelligence. Our study provides insights into how neurally simple limbless organisms like nematodes can leverage mechanical intelligence via appropriately tuned bilateral actuation to locomote in complex environments. These principles likely apply to neurally more sophisticated organisms and also provide a design and control paradigm for limbless robots for applications like search and rescue and planetary exploration.

Cellular computation and cognition

Contemporary neural network models often overlook a central biological fact about neural processing: that single neurons are themselves complex, semi-autonomous computing systems. Both the information processing and information storage abilities of actual biological neurons vastly exceed the simple weighted sum of synaptic inputs computed by the "units" in standard neural network models. Neurons are eukaryotic cells that store information not only in synapses, but also in their dendritic structure and connectivity, as well as genetic "marking" in the epigenome of each individual cell. Each neuron computes a complex nonlinear function of its inputs, roughly equivalent in processing capacity to an entire 1990s-era neural network model. Furthermore, individual cells provide the biological interface between gene expression, ongoing neural processing, and stored long-term memory traces. Neurons in all organisms have these properties, which are thus relevant to all of neuroscience and cognitive biology. Single-cell computation may also play a particular role in explaining some unusual features of human cognition. The recognition of the centrality of cellular computation to "natural computation" in brains, and of the constraints it imposes upon brain evolution, thus has important implications for the evolution of cognition, and how we study it.

The homeodomain transcriptional regulator DVE-1 directs a program for synapse elimination during circuit remodeling

The elimination of synapses during circuit remodeling is critical for brain maturation; however, the molecular mechanisms directing synapse elimination and its timing remain elusive. We show that the transcriptional regulator DVE-1, which shares homology with special AT-rich sequence-binding (SATB) family members previously implicated in human neurodevelopmental disorders, directs the elimination of juvenile synaptic inputs onto remodeling C. elegans GABAergic neurons. Juvenile acetylcholine receptor clusters and apposing presynaptic sites are eliminated during the maturation of wild-type GABAergic neurons but persist into adulthood in dve-1 mutants, producing heightened motor connectivity. DVE-1 localization to GABAergic nuclei is required for synapse elimination, consistent with DVE-1 regulation of transcription. Pathway analysis of putative DVE-1 target genes, proteasome inhibitor, and genetic experiments implicate the ubiquitin-proteasome system in synapse elimination. Together, our findings define a previously unappreciated role for a SATB family member in directing synapse elimination during circuit remodeling, likely through transcriptional regulation of protein degradation processes.

Topographic axonal projection at single-cell precision supports local retinotopy in the mouse superior colliculus

Retinotopy, like all long-range projections, can arise from the axons themselves or their targets. The underlying connectivity pattern, however, remains elusive at the fine scale in the mammalian brain. To address this question, we functionally mapped the spatial organization of the input axons and target neurons in the female mouse retinocollicular pathway at single-cell resolution using in vivo two-photon calcium imaging. We found a near-perfect retinotopic tiling of retinal ganglion cell axon terminals, with an average error below 30 μm or 2° of visual angle. The precision of retinotopy was relatively lower for local neurons in the superior colliculus. Subsequent data-driven modeling ascribed it to a low input convergence, on average 5.5 retinal ganglion cell inputs per postsynaptic cell in the superior colliculus. These results indicate that retinotopy arises largely from topographically precise input from presynaptic cells, rather than elaborating local circuitry to reconstruct the topography by postsynaptic cells.

The neuropeptidergic connectome of C. elegans

Efforts are ongoing to map synaptic wiring diagrams, or connectomes, to understand the neural basis of brain function. However, chemical synapses represent only one type of functionally important neuronal connection; in particular, extrasynaptic, "wireless" signaling by neuropeptides is widespread and plays essential roles in all nervous systems. By integrating single-cell anatomical and gene-expression datasets with biochemical analysis of receptor-ligand interactions, we have generated a draft connectome of neuropeptide signaling in the C. elegans nervous system. This network is characterized by high connection density, extended signaling cascades, autocrine foci, and a decentralized topology, with a large, highly interconnected core containing three constituent communities sharing similar patterns of input connectivity. Intriguingly, several key network hubs are little-studied neurons that appear specialized for peptidergic neuromodulation. We anticipate that the C. elegans neuropeptidergic connectome will serve as a prototype to understand how networks of neuromodulatory signaling are organized.

A complete reconstruction of the early visual system of an adult insect

For most model organisms in neuroscience, research into visual processing in the brain is difficult because of a lack of high-resolution maps that capture complex neuronal circuitry. The microinsect Megaphragma viggianii, because of its small size and non-trivial behavior, provides a unique opportunity for tractable whole-organism connectomics. We image its whole head using serial electron microscopy. We reconstruct its compound eye and analyze the optical properties of the ommatidia as well as the connectome of the first visual neuropil-the lamina. Compared with the fruit fly and the honeybee, Megaphragma visual system is highly simplified: it has 29 ommatidia per eye and 6 lamina neuron types. We report features that are both stereotypical among most ommatidia and specialized to some. By identifying the "barebones" circuits critical for flying insects, our results will facilitate constructing computational models of visual processing in insects.

Why an animal needs a brain

In Principles of Neural Design (2015, MIT Press), inspired by Charles Darwin, Sterling and Laughlin undertook the unfashionable task of distilling principles from facts in the technique-driven, data-saturated domain of neuroscience. Their starting point for deriving the organizing principles of brains are two brainless single-celled organisms, Escherichia coli and Paramecium, and the 302-neuron brain of the nematode Caenorhabditis elegans. The book is an exemplar in how to connect the dots between simpler and (much) more complex organisms in a particular area. Here, they have generously agreed to republish an abridged version of Chapter 2 (Why an Animal Needs a Brain), in which many of their principles are first described.

Chemical cognition: chemoconnectomics and convergent evolution of integrative systems in animals

Neurons underpin cognition in animals. However, the roots of animal cognition are elusive from both mechanistic and evolutionary standpoints. Two conceptual frameworks both highlight and promise to address these challenges. First, we discuss evidence that animal neural and other integrative systems evolved more than once (convergent evolution) within basal metazoan lineages, giving us unique experiments by Nature for future studies. The most remarkable examples are neural systems in ctenophores and neuroid-like systems in placozoans and sponges. Second, in addition to classical synaptic wiring, a chemical connectome mediated by hundreds of signal molecules operates in tandem with neurons and is the most information-rich source of emerging properties and adaptability. The major gap-dynamic, multifunctional chemical micro-environments in nervous systems-is not understood well. Thus, novel tools and information are needed to establish mechanistic links between orchestrated, yet cell-specific, volume transmission and behaviors. Uniting what we call chemoconnectomics and analyses of the cellular bases of behavior in basal metazoan lineages arguably would form the foundation for deciphering the origins and early evolution of elementary cognition and intelligence.

Brain-wide representations of behavior spanning multiple timescales and states in C. elegans

Changes in an animal's behavior and internal state are accompanied by widespread changes in activity across its brain. However, how neurons across the brain encode behavior and how this is impacted by state is poorly understood. We recorded brain-wide activity and the diverse motor programs of freely moving C. elegans and built probabilistic models that explain how each neuron encodes quantitative behavioral features. By determining the identities of the recorded neurons, we created an atlas of how the defined neuron classes in the C. elegans connectome encode behavior. Many neuron classes have conjunctive representations of multiple behaviors. Moreover, although many neurons encode current motor actions, others integrate recent actions. Changes in behavioral state are accompanied by widespread changes in how neurons encode behavior, and we identify these flexible nodes in the connectome. Our results provide a global map of how the cell types across an animal's brain encode its behavior.

Brain network communication: concepts, models and applications

Understanding communication and information processing in nervous systems is a central goal of neuroscience. Over the past two decades, advances in connectomics and network neuroscience have opened new avenues for investigating polysynaptic communication in complex brain networks. Recent work has brought into question the mainstay assumption that connectome signalling occurs exclusively via shortest paths, resulting in a sprawling constellation of alternative network communication models. This Review surveys the latest developments in models of brain network communication. We begin by drawing a conceptual link between the mathematics of graph theory and biological aspects of neural signalling such as transmission delays and metabolic cost. We organize key network communication models and measures into a taxonomy, aimed at helping researchers navigate the growing number of concepts and methods in the literature. The taxonomy highlights the pros, cons and interpretations of different conceptualizations of connectome signalling. We showcase the utility of network communication models as a flexible, interpretable and tractable framework to study brain function by reviewing prominent applications in basic, cognitive and clinical neurosciences. Finally, we provide recommendations to guide the future development, application and validation of network communication models.

The Intrinsic Similarity of Topological Structure in Biological Neural Networks

Most previous studies mainly have focused on the analysis of structural properties of individual neuronal networks from C. elegans. In recent years, an increasing number of synapse-level neural maps, also known as biological neural networks, have been reconstructed. However, it is not clear whether there are intrinsic similarities of structural properties of biological neural networks from different brain compartments or species. To explore this issue, we collected nine connectomes at synaptic resolution including C. elegans, and analyzed their structural properties. We found that these biological neural networks possess small-world properties and modules. Excluding the Drosophila larval visual system, these networks have rich clubs. The distributions of synaptic connection strength for these networks can be fitted by the truncated pow-law distributions. Additionally, compared with the power-law model, a log-normal distribution is a better model to fit the complementary cumulative distribution function (CCDF) of degree for these neuronal networks. Moreover, we also observed that these neural networks belong to the same superfamily based on the significance profile (SP) of small subgraphs in the network. Taken together, these findings suggest that biological neural networks share intrinsic similarities in their topological structure, revealing some principles underlying the formation of biological neural networks within and across species.

A molecular atlas of adult C. elegans motor neurons reveals ancient diversity delineated by conserved transcription factor codes

Motor neurons (MNs) constitute an ancient cell type targeted by multiple adult-onset diseases. It is therefore important to define the molecular makeup of adult MNs in animal models and extract organizing principles. Here, we generated a comprehensive molecular atlas of adult Caenorhabditis elegans MNs and a searchable database (http://celegans.spinalcordatlas.org). Single-cell RNA-sequencing of 13,200 cells revealed that ventral nerve cord MNs cluster into 29 molecularly distinct subclasses. All subclasses are delineated by unique expression codes of either neuropeptide or transcription factor gene families. Strikingly, we found that combinatorial codes of homeodomain transcription factor genes define adult MN diversity both in C. elegans and mice. Further, molecularly defined MN subclasses in C. elegans display distinct patterns of connectivity. Hence, our study couples the connectivity map of the C. elegans motor circuit with a molecular atlas of its constituent MNs, and uncovers organizing principles and conserved molecular codes of adult MN diversity.

How to incorporate biological insights into network models and why it matters

Due to the staggering complexity of the brain and its neural circuitry, neuroscientists rely on the analysis of mathematical models to elucidate its function. From Hodgkin and Huxley's detailed description of the action potential in 1952 to today, new theories and increasing computational power have opened up novel avenues to study how neural circuits implement the computations that underlie behaviour. Computational neuroscientists have developed many models of neural circuits that differ in complexity, biological realism or emergent network properties. With recent advances in experimental techniques for detailed anatomical reconstructions or large-scale activity recordings, rich biological data have become more available. The challenge when building network models is to reflect experimental results, either through a high level of detail or by finding an appropriate level of abstraction. Meanwhile, machine learning has facilitated the development of artificial neural networks, which are trained to perform specific tasks. While they have proven successful at achieving task-oriented behaviour, they are often abstract constructs that differ in many features from the physiology of brain circuits. Thus, it is unclear whether the mechanisms underlying computation in biological circuits can be investigated by analysing artificial networks that accomplish the same function but differ in their mechanisms. Here, we argue that building biologically realistic network models is crucial to establishing causal relationships between neurons, synapses, circuits and behaviour. More specifically, we advocate for network models that consider the connectivity structure and the recorded activity dynamics while evaluating task performance.

Whole-brain annotation and multi-connectome cell typing quantifies circuit stereotypy in Drosophila

The fruit fly Drosophila melanogaster combines surprisingly sophisticated behaviour with a highly tractable nervous system. A large part of the fly's success as a model organism in modern neuroscience stems from the concentration of collaboratively generated molecular genetic and digital resources. As presented in our FlyWire companion paper 1 , this now includes the first full brain connectome of an adult animal. Here we report the systematic and hierarchical annotation of this ~130,000-neuron connectome including neuronal classes, cell types and developmental units (hemilineages). This enables any researcher to navigate this huge dataset and find systems and neurons of interest, linked to the literature through the Virtual Fly Brain database 2 . Crucially, this resource includes 4,552 cell types. 3,094 are rigorous consensus validations of cell types previously proposed in the hemibrain connectome 3 . In addition, we propose 1,458 new cell types, arising mostly from the fact that the FlyWire connectome spans the whole brain, whereas the hemibrain derives from a subvolume. Comparison of FlyWire and the hemibrain showed that cell type counts and strong connections were largely stable, but connection weights were surprisingly variable within and across animals. Further analysis defined simple heuristics for connectome interpretation: connections stronger than 10 unitary synapses or providing >1% of the input to a target cell are highly conserved. Some cell types showed increased variability across connectomes: the most common cell type in the mushroom body, required for learning and memory, is almost twice as numerous in FlyWire as the hemibrain. We find evidence for functional homeostasis through adjustments of the absolute amount of excitatory input while maintaining the excitation-inhibition ratio. Finally, and surprisingly, about one third of the cell types proposed in the hemibrain connectome could not yet be reliably identified in the FlyWire connectome. We therefore suggest that cell types should be defined to be robust to inter-individual variation, namely as groups of cells that are quantitatively more similar to cells in a different brain than to any other cell in the same brain. Joint analysis of the FlyWire and hemibrain connectomes demonstrates the viability and utility of this new definition. Our work defines a consensus cell type atlas for the fly brain and provides both an intellectual framework and open source toolchain for brain-scale comparative connectomics.

Neuronal wiring diagram of an adult brain

Connections between neurons can be mapped by acquiring and analyzing electron microscopic (EM) brain images. In recent years, this approach has been applied to chunks of brains to reconstruct local connectivity maps that are highly informative, yet inadequate for understanding brain function more globally. Here, we present the first neuronal wiring diagram of a whole adult brain, containing 5×107 chemical synapses between ~130,000 neurons reconstructed from a female Drosophila melanogaster. The resource also incorporates annotations of cell classes and types, nerves, hemilineages, and predictions of neurotransmitter identities. Data products are available by download, programmatic access, and interactive browsing and made interoperable with other fly data resources. We show how to derive a projectome, a map of projections between regions, from the connectome. We demonstrate the tracing of synaptic pathways and the analysis of information flow from inputs (sensory and ascending neurons) to outputs (motor, endocrine, and descending neurons), across both hemispheres, and between the central brain and the optic lobes. Tracing from a subset of photoreceptors all the way to descending motor pathways illustrates how structure can uncover putative circuit mechanisms underlying sensorimotor behaviors. The technologies and open ecosystem of the FlyWire Consortium set the stage for future large-scale connectome projects in other species.

Embodied cognitive morphogenesis as a route to intelligent systems

The embryological view of development is that coordinated gene expression, cellular physics and migration provides the basis for phenotypic complexity. This stands in contrast with the prevailing view of embodied cognition, which claims that informational feedback between organisms and their environment is key to the emergence of intelligent behaviours. We aim to unite these two perspectives as embodied cognitive morphogenesis, in which morphogenetic symmetry breaking produces specialized organismal subsystems which serve as a substrate for the emergence of autonomous behaviours. As embodied cognitive morphogenesis produces fluctuating phenotypic asymmetry and the emergence of information processing subsystems, we observe three distinct properties: acquisition, generativity and transformation. Using a generic organismal agent, such properties are captured through models such as tensegrity networks, differentiation trees and embodied hypernetworks, providing a means to identify the context of various symmetry-breaking events in developmental time. Related concepts that help us define this phenotype further include concepts such as modularity, homeostasis and 4E (embodied, enactive, embedded and extended) cognition. We conclude by considering these autonomous developmental systems as a process called connectogenesis, connecting various parts of the emerged phenotype into an approach useful for the analysis of organisms and the design of bioinspired computational agents.

Cyclic structure with cellular precision in a vertebrate sensorimotor neural circuit

Neuronal wiring diagrams reconstructed by electron microscopy1,2,3,4,5 pose new questions about the organization of nervous systems following the time-honored tradition of cross-species comparisons.6,7 The C. elegans connectome has been conceptualized as a sensorimotor circuit that is approximately feedforward,8,9,10,11 starting from sensory neurons proceeding to interneurons and ending with motor neurons. Overrepresentation of a 3-cell motif often known as the "feedforward loop" has provided further evidence for feedforwardness.10,12 Here, we contrast with another sensorimotor wiring diagram that was recently reconstructed from a larval zebrafish brainstem.13 We show that the 3-cycle, another 3-cell motif, is highly overrepresented in the oculomotor module of this wiring diagram. This is a first for any neuronal wiring diagram reconstructed by electron microscopy, whether invertebrate12,14 or mammalian.15,16,17 The 3-cycle of cells is "aligned" with a 3-cycle of neuronal groups in a stochastic block model (SBM)18 of the oculomotor module. However, the cellular cycles exhibit more specificity than can be explained by the group cycles-recurrence to the same neuron is surprisingly common. Cyclic structure could be relevant for theories of oculomotor function that depend on recurrent connectivity. The cyclic structure coexists with the classic vestibulo-ocular reflex arc for horizontal eye movements,19 and could be relevant for recurrent network models of temporal integration by the oculomotor system.20,21.

Connecting Connectomes to Physiology

With the advent of volumetric EM techniques, large connectomic datasets are being created, providing neuroscience researchers with knowledge about the full connectivity of neural circuits under study. This allows for numerical simulation of detailed, biophysical models of each neuron participating in the circuit. However, these models typically include a large number of parameters, and insight into which of these are essential for circuit function is not readily obtained. Here, we review two mathematical strategies for gaining insight into connectomics data: linear dynamical systems analysis and matrix reordering techniques. Such analytical treatment can allow us to make predictions about time constants of information processing and functional subunits in large networks.SIGNIFICANCE STATEMENT This viewpoint provides a concise overview on how to extract important insights from Connectomics data by mathematical methods. First, it explains how new dynamics and new time constants can evolve, simply through connectivity between neurons. These new time-constants can be far longer than the intrinsic membrane time-constants of the individual neurons. Second, it summarizes how structural motifs in the circuit can be discovered. Specifically, there are tools to decide whether or not a circuit is strictly feed-forward or whether feed-back connections exist. Only by reordering connectivity matrices can such motifs be made visible.

Linking cortex and contraction-Integrating models along the corticomuscular pathway

Computational models of the neuromusculoskeletal system provide a deterministic approach to investigate input-output relationships in the human motor system. Neuromusculoskeletal models are typically used to estimate muscle activations and forces that are consistent with observed motion under healthy and pathological conditions. However, many movement pathologies originate in the brain, including stroke, cerebral palsy, and Parkinson's disease, while most neuromusculoskeletal models deal exclusively with the peripheral nervous system and do not incorporate models of the motor cortex, cerebellum, or spinal cord. An integrated understanding of motor control is necessary to reveal underlying neural-input and motor-output relationships. To facilitate the development of integrated corticomuscular motor pathway models, we provide an overview of the neuromusculoskeletal modelling landscape with a focus on integrating computational models of the motor cortex, spinal cord circuitry, α-motoneurons  and skeletal muscle in regard to their role in generating voluntary muscle contraction. Further, we highlight the challenges and opportunities associated with an integrated corticomuscular pathway model, such as challenges in defining neuron connectivities, modelling standardisation, and opportunities in applying models to study emergent behaviour. Integrated corticomuscular pathway models have applications in brain-machine-interaction, education, and our understanding of neurological disease.

Inhibitory neurons control the consolidation of neural assemblies via adaptation to selective stimuli

Brain circuits display modular architecture at different scales of organization. Such neural assemblies are typically associated to functional specialization but the mechanisms leading to their emergence and consolidation still remain elusive. In this paper we investigate the role of inhibition in structuring new neural assemblies driven by the entrainment to various inputs. In particular, we focus on the role of partially synchronized dynamics for the creation and maintenance of structural modules in neural circuits by considering a network of excitatory and inhibitory [Formula: see text]-neurons with plastic Hebbian synapses. The learning process consists of an entrainment to temporally alternating stimuli that are applied to separate regions of the network. This entrainment leads to the emergence of modular structures. Contrary to common practice in artificial neural networks-where the acquired weights are typically frozen after the learning session-we allow for synaptic adaptation even after the learning phase. We find that the presence of inhibitory neurons in the network is crucial for the emergence and the post-learning consolidation of the modular structures. Indeed networks made of purely excitatory neurons or of neurons not respecting Dale's principle are unable to form or to maintain the modular architecture induced by the stimuli. We also demonstrate that the number of inhibitory neurons in the network is directly related to the maximal number of neural assemblies that can be consolidated, supporting the idea that inhibition has a direct impact on the memory capacity of the neural network.

The Caenorhabditis elegans innexin INX-20 regulates nociceptive behavioral sensitivity

Organisms rely on chemical cues in their environment to indicate the presence or absence of food, reproductive partners, predators, or other harmful stimuli. In the nematode Caenorhabditis elegans, the bilaterally symmetric pair of ASH sensory neurons serves as the primary nociceptors. ASH activation by aversive stimuli leads to backward locomotion and stimulus avoidance. We previously reported a role for guanylyl cyclases in dampening nociceptive sensitivity that requires an innexin-based gap junction network to pass cGMP between neurons. Here, we report that animals lacking function of the gap junction component INX-20 are hypersensitive in their behavioral response to both soluble and volatile chemical stimuli that signal through G protein-coupled receptor pathways in ASH. We find that expressing inx-20 in the ADL and AFD sensory neurons is sufficient to dampen ASH sensitivity, which is supported by new expression analysis of endogenous INX-20 tagged with mCherry via the CRISPR-Cas9 system. Although ADL does not form gap junctions directly with ASH, it does so via gap junctions with the interneuron RMG and the sensory neuron ASK. Ablating either ADL or RMG and ASK also resulted in nociceptive hypersensitivity, suggesting an important role for RMG/ASK downstream of ADL in the ASH modulatory circuit. This work adds to our growing understanding of the repertoire of ways by which ASH activity is regulated via its connectivity to other neurons and identifies a previously unknown role for ADL and RMG in the modulation of aversive behavior.

Graph Motif Entropy for Understanding Time-Evolving Networks

The structure of networks can be efficiently represented using motifs, which are those subgraphs that recur most frequently. One route to understanding the motif structure of a network is to study the distribution of subgraphs using statistical mechanics. In this article, we address the use of motifs as network primitives using the cluster expansion from statistical physics. By mapping the network motifs to clusters in the gas model, we derive the partition function for a network, and this allows us to calculate global thermodynamic quantities, such as energy and entropy. We present analytical expressions for the number of certain types of motifs, and compute their associated entropy. We conduct numerical experiments for synthetic and real-world data sets and evaluate the qualitative and quantitative characterizations of the motif entropy derived from the partition function. We find that the motif entropy for real-world networks, such as financial stock market networks, is sensitive to the variance in network structure. This is in line with recent evidence that network motifs can be regarded as basic elements with well-defined information-processing functions.

Network inference from short, noisy, low time-resolution, partial measurements: Application to C. elegans neuronal calcium dynamics

Network link inference from measured time series data of the behavior of dynamically interacting network nodes is an important problem with wide-ranging applications, e.g., estimating synaptic connectivity among neurons from measurements of their calcium fluorescence. Network inference methods typically begin by using the measured time series to assign to any given ordered pair of nodes a numerical score reflecting the likelihood of a directed link between those two nodes. In typical cases, the measured time series data may be subject to limitations, including limited duration, low sampling rate, observational noise, and partial nodal state measurement. However, it is unknown how the performance of link inference techniques on such datasets depends on these experimental limitations of data acquisition. Here, we utilize both synthetic data generated from coupled chaotic systems as well as experimental data obtained from Caenorhabditis elegans neural activity to systematically assess the influence of data limitations on the character of scores reflecting the likelihood of a directed link between a given node pair. We do this for three network inference techniques: Granger causality, transfer entropy, and, a machine learning-based method. Furthermore, we assess the ability of appropriate surrogate data to determine statistical confidence levels associated with the results of link-inference techniques.

The connectome of an insect brain

Brains contain networks of interconnected neurons and so knowing the network architecture is essential for understanding brain function. We therefore mapped the synaptic-resolution connectome of an entire insect brain (Drosophila larva) with rich behavior, including learning, value computation, and action selection, comprising 3016 neurons and 548,000 synapses. We characterized neuron types, hubs, feedforward and feedback pathways, as well as cross-hemisphere and brain-nerve cord interactions. We found pervasive multisensory and interhemispheric integration, highly recurrent architecture, abundant feedback from descending neurons, and multiple novel circuit motifs. The brain's most recurrent circuits comprised the input and output neurons of the learning center. Some structural features, including multilayer shortcuts and nested recurrent loops, resembled state-of-the-art deep learning architectures. The identified brain architecture provides a basis for future experimental and theoretical studies of neural circuits.

Computational Models of Typical and Atypical Brain Network Development

Over the last decade, the organization of brain networks at both micro- and macroscales has become a key focus of neuroscientific inquiry. This has revealed fundamental features of brain network organization-small-worldness, modularity, heavy-tailed degree distributions-and has highlighted how these structural features support brain function. However, the driving forces that shape brain networks over the course of development have begun to be explored only recently. Here, we review recent efforts to gain insights into the mechanisms of brain development through generative modeling of both macroscale human brain networks and microscale cellular connectomes in Caenorhabditis elegans and other organisms. We show how these mathematical models can begin to shed light on the biological processes that drive and constrain the development of brain networks. Finally, we show how generative network models can translate genetic and environmental differences into variability in developmental trajectories, leading to diverse cognitive and mental health outcomes in children and young people.

Analyzing bursting synchronization in structural connectivity matrix of a human brain under external pulsed currents

Cognitive tasks in the human brain are performed by various cortical areas located in the cerebral cortex. The cerebral cortex is separated into different areas in the right and left hemispheres. We consider one human cerebral cortex according to a network composed of coupled subnetworks with small-world properties. We study the burst synchronization and desynchronization in a human neuronal network under external periodic and random pulsed currents. With and without external perturbations, the emergence of bursting synchronization is observed. Synchronization can contribute to the processing of information, however, there are evidences that it can be related to some neurological disorders. Our results show that synchronous behavior can be suppressed by means of external pulsed currents.

The synaptic organization in the Caenorhabditis elegans neural network suggests significant local compartmentalized computations

Neurons are characterized by elaborate tree-like dendritic structures that support local computations by integrating multiple inputs from upstream presynaptic neurons. It is less clear whether simple neurons, consisting of a few or even a single neurite, may perform local computations as well. To address this question, we focused on the compact neural network of Caenorhabditis elegans animals for which the full wiring diagram is available, including the coordinates of individual synapses. We find that the positions of the chemical synapses along the neurites are not randomly distributed nor can they be explained by anatomical constraints. Instead, synapses tend to form clusters, an organization that supports local compartmentalized computations. In mutually synapsing neurons, connections of opposite polarity cluster separately, suggesting that positive and negative feedback dynamics may be implemented in discrete compartmentalized regions along neurites. In triple-neuron circuits, the nonrandom synaptic organization may facilitate local functional roles, such as signal integration and coordinated activation of functionally related downstream neurons. These clustered synaptic topologies emerge as a guiding principle in the network, presumably to facilitate distinct parallel functions along a single neurite, which effectively increase the computational capacity of the neural network.

Learning the dynamics of realistic models of C. elegans nervous system with recurrent neural networks

Given the inherent complexity of the human nervous system, insight into the dynamics of brain activity can be gained from studying smaller and simpler organisms. While some of the potential target organisms are simple enough that their behavioural and structural biology might be well-known and understood, others might still lead to computationally intractable models that require extensive resources to simulate. Since such organisms are frequently only acting as proxies to further our understanding of underlying phenomena or functionality, often one is not interested in the detailed evolution of every single neuron in the system. Instead, it is sufficient to observe the subset of neurons that capture the effect that the profound nonlinearities of the neuronal system have in response to different stimuli. In this paper, we consider the well-known nematode Caenorhabditis elegans and seek to investigate the possibility of generating lower complexity models that capture the system's dynamics with low error using only measured or simulated input-output information. Such models are often termed black-box models. We show how the nervous system of C. elegans can be modelled and simulated with data-driven models using different neural network architectures. Specifically, we target the use of state-of-the-art recurrent neural network architectures such as Long Short-Term Memory and Gated Recurrent Units and compare these architectures in terms of their properties and their accuracy (Root Mean Square Error), as well as the complexity of the resulting models. We show that Gated Recurrent Unit models with a hidden layer size of 4 are able to accurately reproduce the system response to very different stimuli. We furthermore explore the relative importance of their inputs as well as scalability to more scenarios.

Emergent dynamics in a robotic model based on the Caenorhabditis elegans connectome

We analyze the neural dynamics and their relation with the emergent actions of a robotic vehicle that is controlled by a neural network numerical simulation based on the nervous system of the nematode Caenorhabditis elegans. The robot interacts with the environment through a sensor that transmits the information to sensory neurons, while motor neurons outputs are connected to wheels. This is enough to allow emergent robot actions in complex environments, such as avoiding collisions with obstacles. Working with robotic models makes it possible to simultaneously keep track of the dynamics of all the neurons and also register the actions of the robot in the environment in real time, while avoiding the complex technicalities of simulating a real environment. This allowed us to identify several relevant features of the neural dynamics associated with the emergent actions of the robot, some of which have already been observed in biological worms. These results suggest that some basic aspects of behaviors observed in living beings are determined by the underlying structure of the associated neural network.

Quantitative description of neuronal calcium dynamics in C. elegans' thermoreception

The dynamical mechanisms underlying thermoreception in the nematode C. elegans are studied with a mathematical model for the amphid finger-like ciliated (AFD) neurons. The equations, equipped with Arrhenius temperature factors, account for the worm's thermotaxis when seeking environments at its cultivation temperature, and for the AFD's calcium dynamics when exposed to both linearly ramping and oscillatory temperature stimuli. Calculations of the peak time for calcium responses during simulations of pulse-like temperature inputs are consistent with known behavioral time scales of C. elegans.

Polymer Physics-Based Classification of Neurons

Recognizing that diverse morphologies of neurons are reminiscent of structures of branched polymers, we put forward a principled and systematic way of classifying neurons that employs the ideas of polymer physics. In particular, we use 3D coordinates of individual neurons, which are accessible in recent neuron reconstruction datasets from electron microscope images. We numerically calculate the form factor, F(q), a Fourier transform of the distance distribution of particles comprising an object of interest, which is routinely measured in scattering experiments to quantitatively characterize the structure of materials. For a polymer-like object consisting of n monomers spanning over a length scale of r, F(q) scales with the wavenumber [Formula: see text] as [Formula: see text] at an intermediate range of q, where [Formula: see text] is the fractal dimension or the inverse scaling exponent ([Formula: see text]) characterizing the geometrical feature ([Formula: see text]) of the object. F(q) can be used to describe a neuron morphology in terms of its size ([Formula: see text]) and the extent of branching quantified by [Formula: see text]. By defining the distance between F(q)s as a measure of similarity between two neuronal morphologies, we tackle the neuron classification problem. In comparison with other existing classification methods for neuronal morphologies, our F(q)-based classification rests solely on 3D coordinates of neurons with no prior knowledge of morphological features. When applied to publicly available neuron datasets from three different organisms, our method not only complements other methods but also offers a physical picture of how the dendritic and axonal branches of an individual neuron fill the space of dense neural networks inside the brain.

Modeling the Evolution of Biological Neural Networks Based on Caenorhabditis elegans Connectomes across Development

Knowledge of the structural properties of biological neural networks can help in understanding how particular responses and actions are generated. Recently, Witvliet et al. published the connectomes of eight isogenic Caenorhabditis elegans hermaphrodites at different postembryonic ages, from birth to adulthood. We analyzed the basic structural properties of these biological neural networks. From birth to adulthood, the asymmetry between in-degrees and out-degrees over the C. elegans neuronal network increased with age, in addition to an increase in the number of nodes and edges. The degree distributions were neither Poisson distributions nor pure power-law distributions. We have proposed a model of network evolution with different initial attractiveness for in-degrees and out-degrees of nodes and preferential attachment, which reproduces the asymmetry between in-degrees and out-degrees and similar degree distributions via the tuning of the initial attractiveness values. In this study, we present the well-preserved structural properties of C. elegans neuronal networks across development, and provide some insight into understanding the evolutionary processes of biological neural networks through a simple network model.

Alternative neural systems: What is a neuron? (Ctenophores, sponges and placozoans)

How to make a neuron, a synapse, and a neural circuit? Is there only one 'design' for a neural architecture with a universally shared genomic blueprint across species? The brief answer is "No." Four early divergent lineages from the nerveless common ancestor of all animals independently evolved distinct neuroid-type integrative systems. One of these is a subset of neural nets in comb jellies with unique synapses; the second lineage is the well-known Cnidaria + Bilateria; the two others are non-synaptic neuroid systems in sponges and placozoans. By integrating scRNA-seq and microscopy data, we revise the definition of neurons as synaptically-coupled polarized and highly heterogenous secretory cells at the top of behavioral hierarchies with learning capabilities. This physiological (not phylogenetic) definition separates 'true' neurons from non-synaptically and gap junction-coupled integrative systems executing more stereotyped behaviors. Growing evidence supports the hypothesis of multiple origins of neurons and synapses. Thus, many non-bilaterian and bilaterian neuronal classes, circuits or systems are considered functional rather than genetic categories, composed of non-homologous cell types. In summary, little-explored examples of convergent neuronal evolution in representatives of early branching metazoans provide conceptually novel microanatomical and physiological architectures of behavioral controls in animals with prospects of neuro-engineering and synthetic biology.

Chaotic heteroclinic networks as models of switching behavior in biological systems

Key features of biological activity can often be captured by transitions between a finite number of semi-stable states that correspond to behaviors or decisions. We present here a broad class of dynamical systems that are ideal for modeling such activity. The models we propose are chaotic heteroclinic networks with nontrivial intersections of stable and unstable manifolds. Due to the sensitive dependence on initial conditions, transitions between states are seemingly random. Dwell times, exit distributions, and other transition statistics can be built into the model through geometric design and can be controlled by tunable parameters. To test our model's ability to simulate realistic biological phenomena, we turned to one of the most studied organisms, C. elegans, well known for its limited behavioral states. We reconstructed experimental data from two laboratories, demonstrating the model's ability to quantitatively reproduce dwell times and transition statistics under a variety of conditions. Stochastic switching between dominant states in complex dynamical systems has been extensively studied and is often modeled as Markov chains. As an alternative, we propose here a new paradigm, namely, chaotic heteroclinic networks generated by deterministic rules (without the necessity for noise). Chaotic heteroclinic networks can be used to model systems with arbitrary architecture and size without a commensurate increase in phase dimension. They are highly flexible and able to capture a wide range of transition characteristics that can be adjusted through control parameters.

Modulating cell signalling in vivo with magnetic nanotransducers

Weak magnetic fields offer nearly lossless transmission of signals within biological tissue. Magnetic nanomaterials are capable of transducing magnetic fields into a range of biologically relevant signals in vitro and in vivo. These nanotransducers have recently enabled magnetic control of cellular processes, from neuronal firing and gene expression to programmed apoptosis. Effective implementation of magnetically controlled cellular signalling relies on careful tailoring of magnetic nanotransducers and magnetic fields to the responses of the intended molecular targets. This primer discusses the versatility of magnetic modulation modalities and offers practical guidelines for selection of appropriate materials and field parameters, with a particular focus on applications in neuroscience. With recent developments in magnetic instrumentation and nanoparticle chemistries, including those that are commercially available, magnetic approaches promise to empower research aimed at connecting molecular and cellular signalling to physiology and behaviour in untethered moving subjects.

Identifying accurate link predictors based on assortativity of complex networks

Link prediction algorithms in complex networks, such as social networks, biological networks, drug-drug interactions, communication networks, and so on, assign scores to predict potential links between two nodes. Link prediction (LP) enables researchers to learn unknown, new as well as future interactions among the entities being modeled in the complex networks. In addition to measures like degree distribution, clustering coefficient, centrality, etc., another metric to characterize structural properties is network assortativity which measures the tendency of nodes to connect with similar nodes. In this paper, we explore metrics that effectively predict the links based on the assortativity profiles of the complex networks. To this end, we first propose an approach that generates networks of varying assortativity levels and utilize three sets of link prediction models combining the similarity of neighborhoods and preferential attachment. We carry out experiments to study the LP accuracy (measured in terms of area under the precision-recall curve) of the link predictors individually and in combination with other baseline measures. Our analysis shows that link prediction models that explore a large neighborhood around nodes of interest, such as CH2-L2 and CH2-L3, perform consistently for assortative as well as disassortative networks. While common neighbor-based local measures are effective for assortative networks, our proposed combination of common neighbors with node degree is a good choice for the LP metric in disassortative networks. We discuss how this analysis helps achieve the best-parameterized combination of link prediction models and its significance in the context of link prediction from incomplete social and biological network data.

Explainable artificial intelligence through graph theory by generalized social network analysis-based classifier

We propose a new type of supervised visual machine learning classifier, GSNAc, based on graph theory and social network analysis techniques. In a previous study, we employed social network analysis techniques and introduced a novel classification model (called Social Network Analysis-based Classifier—SNAc) which efficiently works with time-series numerical datasets. In this study, we have extended SNAc to work with any type of tabular data by showing its classification efficiency on a broader collection of datasets that may contain numerical and categorical features. This version of GSNAc simply works by transforming traditional tabular data into a network where samples of the tabular dataset are represented as nodes and similarities between the samples are reflected as edges connecting the corresponding nodes. The raw network graph is further simplified and enriched by its edge space to extract a visualizable ‘graph classifier model—GCM’. The concept of the GSNAc classification model relies on the study of node similarities over network graphs. In the prediction step, the GSNAc model maps test nodes into GCM, and evaluates their average similarity to classes by employing vectorial and topological metrics. The novel side of this research lies in transforming multidimensional data into a 2D visualizable domain. This is realized by converting a conventional dataset into a network of ‘samples’ and predicting classes after a careful and detailed network analysis. We exhibit the classification performance of GSNAc as an effective classifier by comparing it with several well-established machine learning classifiers using some popular benchmark datasets. GSNAc has demonstrated superior or comparable performance compared to other classifiers. Additionally, it introduces a visually comprehensible process for the benefit of end-users. As a result, the spin-off contribution of GSNAc lies in the interpretability of the prediction task since the process is human-comprehensible; and it is highly visual.

Reprogramming the topology of the nociceptive circuit in C. elegans reshapes sexual behavior

The effect of the detailed connectivity of a neural circuit on its function and the resulting behavior of the organism is a key question in many neural systems. Here, we study the circuit for nociception in C. elegans, which is composed of the same neurons in the two sexes that are wired differently. We show that the nociceptive sensory neurons respond similarly in the two sexes, yet the animals display sexually dimorphic behaviors to the same aversive stimuli. To uncover the role of the downstream network topology in shaping behavior, we learn and simulate network models that replicate the observed dimorphic behaviors and use them to predict simple network rewirings that would switch behavior between the sexes. We then show experimentally that these subtle synaptic rewirings indeed flip behavior. Interestingly, when presented with aversive cues, rewired males were compromised in finding mating partners, suggesting that network topologies that enable efficient avoidance of noxious cues have a reproductive "cost." Our results present a deconstruction of the design of a neural circuit that controls sexual behavior and how to reprogram it.

Reduced Ca2+ transient amplitudes may signify increased or decreased depolarization depending on the neuromodulatory signaling pathway

Neuromodulators regulate neuronal excitability and bias neural circuit outputs. Optical recording of neuronal Ca²⁺ transients is a powerful approach to study the impact of neuromodulators on neural circuit dynamics. We are investigating the polymodal nociceptor ASH in Caenorhabditis elegans to better understand the relationship between neuronal excitability and optically recorded Ca²⁺ transients. ASHs depolarize in response to the aversive olfactory stimulus 1-octanol (1-oct) with a concomitant rise in somal Ca²⁺, stimulating an aversive locomotory response. Serotonin (5-HT) potentiates 1-oct avoidance through Gα_q signaling, which inhibits L-type voltage-gated Ca²⁺ channels in ASH. Although Ca²⁺ signals in the ASH soma decrease, depolarization amplitudes increase because Ca²⁺ mediates inhibitory feedback control of membrane potential in this context. Here, we investigate octopamine (OA) signaling in ASH to assess whether this negative correlation between somal Ca²⁺ and depolarization amplitudes is a general phenomenon, or characteristic of certain neuromodulatory pathways. Like 5-HT, OA reduces somal Ca²⁺ transient amplitudes in ASH neurons. However, OA antagonizes 5-HT modulation of 1-oct avoidance behavior, suggesting that OA may signal through a different pathway. We further show that the pathway for OA diminution of ASH somal Ca²⁺ consists of the OCTR-1 receptor, the G_o heterotrimeric G-protein, and the G-protein activated inwardly rectifying channels IRK-2 and IRK-3, and this pathway reduces depolarization amplitudes in parallel with somal Ca²⁺ transient amplitudes. Therefore, even within a single neuron, somal Ca²⁺ signal reduction may indicate either increased or decreased depolarization amplitude, depending on which neuromodulatory signaling pathways are activated, underscoring the need for careful interpretation of Ca²⁺ imaging data in neuromodulatory studies.

Edge-colored directed subgraph enumeration on the connectome

Following significant advances in image acquisition, synapse detection, and neuronal segmentation in connectomics, researchers have extracted an increasingly diverse set of wiring diagrams from brain tissue. Neuroscientists frequently represent these wiring diagrams as graphs with nodes corresponding to a single neuron and edges indicating synaptic connectivity. The edges can contain "colors" or "labels", indicating excitatory versus inhibitory connections, among other things. By representing the wiring diagram as a graph, we can begin to identify motifs, the frequently occurring subgraphs that correspond to specific biological functions. Most analyses on these wiring diagrams have focused on hypothesized motifs-those we expect to find. However, one of the goals of connectomics is to identify biologically-significant motifs that we did not previously hypothesize. To identify these structures, we need large-scale subgraph enumeration to find the frequencies of all unique motifs. Exact subgraph enumeration is a computationally expensive task, particularly in the edge-dense wiring diagrams. Furthermore, most existing methods do not differentiate between types of edges which can significantly affect the function of a motif. We propose a parallel, general-purpose subgraph enumeration strategy to count motifs in the connectome. Next, we introduce a divide-and-conquer community-based subgraph enumeration strategy that allows for enumeration per brain region. Lastly, we allow for differentiation of edges by types to better reflect the underlying biological properties of the graph. We demonstrate our results on eleven connectomes and publish for future analyses extensive overviews for the 26 trillion subgraphs enumerated that required approximately 9.25 years of computation time.

A set of hub neurons and non-local connectivity features support global brain dynamics in C. elegans

The wiring architecture of neuronal networks is assumed to be a strong determinant of their dynamical computations. An ongoing effort in neuroscience is therefore to generate comprehensive synapse-resolution connectomes alongside brain-wide activity maps. However, the structure-function relationship, i.e., how the anatomical connectome and neuronal dynamics relate to each other on a global scale, remains unsolved. Systematically, comparing graph features in the C. elegans connectome with correlations in nervous system-wide neuronal dynamics, we found that few local connectivity motifs and mostly other non-local features such as triplet motifs and input similarities can predict functional relationships between neurons. Surprisingly, quantities such as connection strength and amount of common inputs do not improve these predictions, suggesting that the network's topology is sufficient. We demonstrate that hub neurons in the connectome are key to these relevant graph features. Consistently, inhibition of multiple hub neurons specifically disrupts brain-wide correlations. Thus, we propose that a set of hub neurons and non-local connectivity features provide an anatomical substrate for global brain dynamics.

A Perspective on Information Optimality in a Neural Circuit and Other Biological Systems

The nematode worm Caenorhabditis elegans has a relatively simple neural system for analysis of information transmission from sensory organ to muscle fiber. Consequently, this study includes an example of a neural circuit from the nematode worm, and a procedure is shown for measuring its information optimality by use of a logic gate model. This approach is useful where the assumptions are applicable for a neural circuit, and also for choosing between competing mathematical hypotheses that explain the function of a neural circuit. In this latter case, the logic gate model can estimate computational complexity and distinguish which of the mathematical models require fewer computations. In addition, the concept of information optimality is generalized to other biological systems, along with an extended discussion of its role in genetic-based pathways of organisms.

Neurons as hierarchies of quantum reference frames

Conceptual and mathematical models of neurons have lagged behind empirical understanding for decades. Here we extend previous work in modeling biological systems with fully scale-independent quantum information-theoretic tools to develop a uniform, scalable representation of synapses, dendritic and axonal processes, neurons, and local networks of neurons. In this representation, hierarchies of quantum reference frames act as hierarchical active-inference systems. The resulting model enables specific predictions of correlations between synaptic activity, dendritic remodeling, and trophic reward. We summarize how the model may be generalized to nonneural cells and tissues in developmental and regenerative contexts.